Understanding Sample Rate

[Don Hills]

Aha, I misread the original post. What JJ actually said was "1/ ( 2 pi bandwidth number_of_levels)", not "sample rate". 110 ps, not 55. How embarrassing, I've had it wrong the last 3 years. So he was describing the limit case (22.05 kHz for 16/44.1) whereas your (Mans') derivation in the "MQA is Vaporware" thread is the general derivation.

 

 

 

[Don Hills]

35 minutes ago, mansr said:

Correct, and amplitude matters too. In the formula, the figure for the number of levels should be in relation to the signal peak, which might be smaller than the maximum possible.

 

That would make my earlier calculation 4 bits rather than 3. Still too small to worry about.

[mansr]

32 minutes ago, Don Hills said:

What JJ actually said was "1/ ( 2 pi bandwidth number_of_levels)"

Yes, that is correct.

[adamdea]

42 minutes ago, Don Hills said:

Aha, I misread the original post. What JJ actually said was "1/ ( 2 pi bandwidth number_of_levels)", not "sample rate". 110 ps, not 55. How embarrassing, I've had it wrong the last 3 years. So he was describing the limit case (22.05 kHz for 16/44.1) whereas your (Mans') derivation in the "MQA is Vaporware" thread is the general derivation.

 

 

 

Don, I too share your concern becasue I have assumed JJ got it right. But he definitely said it was 1 

"1/ (2 pi fs nlevels)

for CD that is 1/( 2 pi 44100 65536)."

https://hydrogenaud.io/index.php/topic,91126.0.html

Fs isn't bandwidth. Are we reading the same thing. Sorry if I am being obtuse.

[mansr]

6 minutes ago, adamdea said:

Don, I too share your concern becasue I have assumed JJ got it right. But he definitely said it was

"1/ (2 pi fs nlevels)

for CD that is 1/( 2 pi 44100 65536)."

https://hydrogenaud.io/index.php/topic,91126.0.html

Fs isn't bandwidth. Are we reading the same thing. Sorry if I am being obtuse.

That's definitely wrong.

[adamdea]

Just now, mansr said:

That's definitely wrong.

Just to be clear I assuming that you are saying that what he said was wrong not that I have misquoted him.

[mansr]

Just now, adamdea said:

Just to be clear I assuming that you are saying that what he said was wrong not that I have misquoted him.

He's wrong, the quote is accurate. Maybe he just misspoke, but what he said there is wrong.

[psjug]

9 minutes ago, adamdea said:

Don, I too share your concern becasue I have assumed JJ got it right. But he definitely said it was 1 

"1/ (2 pi fs nlevels)

for CD that is 1/( 2 pi 44100 65536)."

https://hydrogenaud.io/index.php/topic,91126.0.html

Fs isn't bandwidth. Are we reading the same thing. Sorry if I am being obtuse.

Even the smartest people like JJ make mistakes.  

[adamdea]

Great. Can you just talk me through your correction from 220 ps to 110ps

[mansr]

5 minutes ago, adamdea said:

Great. Can you just talk me through your correction from 220 ps to 110ps

Scroll down a little on that page. As esldude points out, a level above ½ LSB will be rounded up to 1.

[adamdea]

5 minutes ago, mansr said:

Scroll down a little on that page. As esldude points out, a level above ½ LSB will be rounded up to 1.

Ah so the minimum time difference is equal to the difference which will cause a 1/2 lsb amplitude change in a 22.05Khz wave at the sampling instant.

 

[beerandmusic]

5 minutes ago, mansr said:

Scroll down a little on that page. As esldude points out, a level above ½ LSB will be rounded up to 1.

rounded up..i thought it could capture perfectly

[esldude]

Also with pointing out that with dither in use you can sample with precision smaller than the LSB.

[mansr]

2 minutes ago, adamdea said:

Ah so the minimum time difference is equal to the difference which will cause a 1/2 lsb amplitude change in a 22.05Khz wave at the sampling instant.

Right. The 110 ps figure is for a full amplitude signal at the highest frequency, 22.05 kHz. This represents the best case. Now look at some other limit cases. For example, a full amplitude sine wave with a very long period, say a year, changes so slowly that it takes more than a minute to go from zero 1 LSB. At another extreme, a sine wave with an amplitude of 1 LSB needs a phase shift of 30° in order for a sample point initially at the zero-crossing to register a 1.

[adamdea]

Thanks 

I'm still a bit troubled. A part of me feel that if a 22.05khz wave is sampled forever and then phase shifted slightly, the minimum time difference to register one sample with a different value (out of all samples) will be different from the figure required to get a difference in the value of one sample per cycle.

Is your calculation not the time difference required to get a 22.05 khz wave perfectly zero crossing at a sample instant to read the lsb at the sampling instant. What trouble me is that any lsb could always tick over to the next value for any change in phase if the value was previously just under the threshold where it rounded up. 

Something nags at my mind that this is  not the question people want to answer when they ask what is the time resolution of the system. In a dithered system the sample values will change anyway, My instinct is that for an infinitely long sampling period we can detect any change in phase eventually.

 

Is the better question what is the minimum change in the location of the peak of a sine wave which can be detected from the sample values? Is that the same as the question you answered?

 

[mansr]

9 minutes ago, adamdea said:

Thanks 

I'm still a bit troubled. A part of me feel that if a 22.05khz wave is sampled forever and then phase shifted slightly, the minimum time difference to register one sample with a different value (out of all samples) will be different from the figure required to get a difference in the value of one sample per cycle.

Is your calculation not the time difference required to get a 22.05 khz wave perfectly zero crossing at a sample instant to read the lsb at the sampling instant. What trouble me is that any lsb could always tick over to the next value for any change in phase if the value was previously just under the threshold where it rounded up. 

Such is the nature of quantisation. A small phase shift will cause a change in some samples but not in others. The reconstructed signal thus gets distorted a little differently in addition to the phase shift.

 

9 minutes ago, adamdea said:

Something nags at my mind that this is  not the question people want to answer when they ask what is the time resolution of the system. In a dithered system the sample values will change anyway, My instinct is that for an infinitely long sampling period we can detect any change in phase eventually.

Disregard dither to begin with. Now consider a sine wave sampled at some multiple of its frequency. In each period of this this wave, the samples values will be exactly the same. If we now shift it slightly, but not enough to trigger an LSB change at any of the sample points, every period will have exactly those same sample values in perpetuity. If the sample rate isn't a multiple of the signal frequency, the sample values will still be periodic, just with a longer period and the same logic applies.

 

9 minutes ago, adamdea said:

Is the better question what is the minimum change in the location of the peak of a sine wave which can be detected from the sample values? Is that the same as the question you answered?

The derivative of the sine is zero at the peaks. To cause an LSB flip there, the phase shift has to be much greater than at the zero-crossing.

[tmtomh]

4 hours ago, beerandmusic said:

rounded up..i thought it could capture perfectly

 

That's bit-depth and volume level/amplitude, aka quantization error, not sample rate and frequency accuracy.

 

A level below LSB for 16-bit PCM sampling would have to be approx. -96dB in volume.

[Ralf11]

12 hours ago, Spacehound said:

The Xiph  video is what you might think of as the "World Standard" And it's not only demonstrated, it's mathematically proven.

 

Anyone who disagrees with it is wrong. It's as simple as that.

 

 

it's even better if you turn on the French subtitles

[Ralf11]

11 hours ago, beerandmusic said:

you understood what i meant....

for your sake, any number between 300 and 301, with any amount of decimal points

i merely take it to extremes for sake of better understanding.

 

 

 

Georg Cantor went insane from thinking about the real line

[bobbmd]

@Spacehound My point exactly... I didn’t bring Pauling into the thread you or someone else did and THAT didn’t have anything to do with the thread either-I was just making a comment/observation humorous sarcastic or otherwise and if you didn't enjoy it so be it have a nice day

[Don Hills]

5 hours ago, adamdea said:

Don, I too share your concern becasue I have assumed JJ got it right. But he definitely said it was 1 

"1/ (2 pi fs nlevels)

for CD that is 1/( 2 pi 44100 65536)."

https://hydrogenaud.io/index.php/topic,91126.0.html

Fs isn't bandwidth. Are we reading the same thing. Sorry if I am being obtuse.

 

That was in 2011. In 2015, he gave the other equation:

https://hydrogenaud.io/index.php/topic,108987.msg896449.html#msg896449

 

So it appears he corrected himself in the meantime. Unlike me, he appears to have retained his faculties with advancing age.

[Ralf11]

8 hours ago, jabbr said:

As @mansr, says, the frequency quantization or time accuracy depends on the sample precision. The sample precision is limited by the baseline noise and can not be arbitrarily reduced by “better electronics” because the uncertainty principle determines the limit of our ability to reduce “noise”.

 

This is no real concern waveforms add in known ways. It all works!

 

do we hit that limit in the real world?

 

I thought Johnson noise, etc. were the practical limits

[fas42]

The precision of digital, in the theoretical sense, is never a problem - all the calculable errors are at the level of noise, which if handled well is random in a way that doesn't annoy the brain. Why the real world systems don't sound quite right is that various implementation and interference issues are not addressed thoroughly enough - and here the maths of what's going on is probably so messy that no-one would want to try and understand it.

 

 

[jabbr]

1 hour ago, Ralf11 said:

 

do we hit that limit in the real world?

 

I thought Johnson noise, etc. were the practical limits

 

At least “shot noise” is described as due to the quantum nature of electrons (https://arxiv.org/pdf/cond-mat/9407011v1.pdf) also 1/f noise (http://physics.princeton.edu/~mcdonald/examples/statistics/handel_pra_22_745_80.pdf)

[Ralf11]

yes, I had added shot noise but then realized my error* and took it out

 

 

* Luckily my errors follow a poisson distribution and go well with a nice meuniere sauce